Probability of a stochastic process crossing a boundary in time interval

Question

Suppose that we have a stochastic process $X(t)$: $$X(t) = \frac{1}{t}\int_{0}^{t} W(\tau) d\tau$$ where $W(\tau)$ is a Wiener process. What is the probability of $X(t)$ crossing a barrier $\alpha$ ($\alpha > 0$, also $\alpha$ is said to be large) at least once in time interval $[t_1, t_2]$?